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      【ARC065E】Manhattan Compass
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        <p>显然是一个BFS，主要问题在于BFS过程中如何快速解决一个点</p>
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<p>根据哈密顿距离所定义的“圆”，是一个每条边都与坐标轴呈45度角的矩形，这让人很头疼</p>
<p>不如我们将它旋转45度，与坐标轴平行，然后哈密顿距离就变成了切比雪夫距离。具体地，对于一个点(x,y)，将其旋转45度并进行适量缩放之后，得到的点为(x+y,x-y)。可以证明，旋转后的点和旋转前的点是唯一对应的</p>
<p>将旋转后点的所有横坐标离散化，然后开若干个set，我们称之为Sx[x]，存储离散化之后横坐标为x的所有点，set里面用pair存储，first域是<strong>未离散的</strong>纵坐标，second域是点的编号。纵坐标也做一样的处理</p>
<p>这样，对于每个点，我们可以算出它矩形的4个顶点，然后算出它们旋转后的坐标。对于每一条边，发现它一定是某个set中的一段连续区间，这样只要在对应的set上二分出这段区间，然后枚举所有点就可以了</p>
<p>但这样会T飞。非常显然，一条边可以存在于若干个矩形中。因为一个已经处理过的点再让他入队是没有意义的，但是在处理不同的点时枚举到它，却是对答案有贡献的，我们需要找出一种不遍历它却又能统计到它对答案贡献的方法</p>
<p>我们开若干个vector，一开始存储的内容和set完全一样。记得存完之后对所有vector排序一下。这样，我们在对应的vector上二分出这段区间，然后区间端点减一下就是对答案的贡献了。set上我们照样枚举，只不过枚举到一个点之后我们把它erase掉，避免后期的重复遍历</p>
<p>注意，矩形顶点若对答案存在贡献，那它在相邻的两条边中都会被统计到答案，需要特殊处理。还有最后答案记得除2</p>
<div class="highlight-box" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false" contenteditable="true" data-rel="CPP"><figure class="iseeu highlight /cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span 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class="line">117</span><br><span class="line">118</span><br><span class="line">119</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> MP make_pair</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> FR first</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> SE second</span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> S=(<span class="number">1</span>&lt;&lt;<span class="number">20</span>)+<span class="number">5</span>;</span><br><span class="line"><span class="keyword">char</span> buf[S],*H,*T;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">char</span> <span class="title">Get</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span>(H==T) T=(H=buf)+fread(buf,<span class="number">1</span>,S,<span class="built_in">stdin</span>);</span><br><span class="line">    <span class="keyword">if</span>(H==T) <span class="keyword">return</span> <span class="number">-1</span>;<span class="keyword">return</span> *H++;</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">read</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> x=<span class="number">0</span>;<span class="keyword">char</span> c=Get();</span><br><span class="line">    <span class="keyword">while</span>(!<span class="built_in">isdigit</span>(c)) c=Get();</span><br><span class="line">    <span class="keyword">while</span>(<span class="built_in">isdigit</span>(c)) x=x*<span class="number">10</span>+c-<span class="string">'0'</span>,c=Get();</span><br><span class="line">    <span class="keyword">return</span> x;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="keyword">typedef</span> pair&lt;<span class="keyword">int</span>,<span class="keyword">int</span>&gt; pii;</span><br><span class="line"><span class="keyword">typedef</span> <span class="keyword">long</span> <span class="keyword">long</span> LL;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N=<span class="number">100010</span>;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Point</span></span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">    <span class="keyword">int</span> x,y;</span><br><span class="line">    Point(<span class="keyword">int</span> _x=<span class="number">0</span>,<span class="keyword">int</span> _y=<span class="number">0</span>):x(_x),y(_y)&#123;&#125;</span><br><span class="line">    <span class="function">Point <span class="title">rotate</span><span class="params">()</span></span>&#123;<span class="keyword">return</span> Point(x+y,y-x);&#125;</span><br><span class="line">    <span class="keyword">bool</span> <span class="keyword">operator</span> != (Point b)&#123;<span class="keyword">return</span> x!=b.x||y!=b.y;&#125;</span><br><span class="line">&#125; org[N],rtg[N],prg[N];</span><br><span class="line"><span class="built_in">set</span>&lt;pii&gt; sx[N],sy[N];</span><br><span class="line"><span class="built_in">vector</span>&lt;pii&gt; vx[N],vy[N];</span><br><span class="line"><span class="built_in">queue</span>&lt;<span class="keyword">int</span>&gt; q;</span><br><span class="line"><span class="keyword">bool</span> vis[N];</span><br><span class="line"><span class="keyword">int</span> Hx[N],Hy[N],mx,my;</span><br><span class="line"><span class="keyword">int</span> n,A,B,d;</span><br><span class="line">LL ans=<span class="number">0</span>;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">prework</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    sort(Hx+<span class="number">1</span>,Hx+<span class="number">1</span>+n);sort(Hy+<span class="number">1</span>,Hy+<span class="number">1</span>+n);</span><br><span class="line">    mx=unique(Hx+<span class="number">1</span>,Hx+<span class="number">1</span>+n)-(Hx+<span class="number">1</span>);</span><br><span class="line">    my=unique(Hy+<span class="number">1</span>,Hy+<span class="number">1</span>+n)-(Hy+<span class="number">1</span>);</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=n;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        prg[i].x=lower_bound(Hx+<span class="number">1</span>,Hx+<span class="number">1</span>+mx,rtg[i].x)-Hx;    <span class="comment">//离散化</span></span><br><span class="line">        prg[i].y=lower_bound(Hy+<span class="number">1</span>,Hy+<span class="number">1</span>+my,rtg[i].y)-Hy;</span><br><span class="line">        sx[prg[i].x].insert(MP(rtg[i].y,i));</span><br><span class="line">        sy[prg[i].y].insert(MP(rtg[i].x,i));</span><br><span class="line">        vx[prg[i].x].push_back(MP(rtg[i].y,i));</span><br><span class="line">        vy[prg[i].y].push_back(MP(rtg[i].x,i));</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=mx;i++) sort(vx[i].begin(),vx[i].end());    <span class="comment">//将所有vector排序</span></span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=my;i++) sort(vy[i].begin(),vy[i].end());</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function">pii <span class="title">gao</span><span class="params">(<span class="built_in">set</span>&lt;pii&gt; &amp;s,<span class="built_in">vector</span>&lt;pii&gt; &amp;v,<span class="keyword">int</span> l,<span class="keyword">int</span> r)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    ans+=upper_bound(v.begin(),v.end(),MP(r,n))-lower_bound(v.begin(),v.end(),MP(l,<span class="number">0</span>));</span><br><span class="line">    <span class="keyword">auto</span> pl=s.lower_bound(MP(l,<span class="number">0</span>)),pr=s.upper_bound(MP(r,n));    <span class="comment">//要注意二分边界问题</span></span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">auto</span> it=pl;it!=pr;)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">if</span>(!vis[it-&gt;SE])</span><br><span class="line">        &#123;</span><br><span class="line">            vis[it-&gt;SE]=<span class="number">1</span>;</span><br><span class="line">            q.push(it-&gt;SE);</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">auto</span> it2=it;it2++;</span><br><span class="line">        s.erase(it);it=it2;    <span class="comment">//将遍历过的元素erase掉</span></span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">auto</span> ppl=lower_bound(v.begin(),v.end(),MP(l,<span class="number">0</span>));</span><br><span class="line">    <span class="keyword">auto</span> ppr=lower_bound(v.begin(),v.end(),MP(r,<span class="number">0</span>));</span><br><span class="line">    <span class="keyword">return</span> MP(ppl==v.end()?<span class="number">0</span>:ppl-&gt;SE,ppr==v.end()?<span class="number">0</span>:ppr-&gt;SE);    <span class="comment">//返回区间端点</span></span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">BFS</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    vis[A]=<span class="number">1</span>;q.push(A);</span><br><span class="line">    <span class="keyword">while</span>(!q.empty())</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">int</span> u=q.front();q.pop();</span><br><span class="line">        pii p1(0,0),p2(0,0),p3(0,0),p4(0,0);</span><br><span class="line">        Point pt=org[u];</span><br><span class="line">        Point pt1=Point(pt.x+d,pt.y).rotate();    <span class="comment">//计算矩形顶点</span></span><br><span class="line">        Point pt2=Point(pt.x,pt.y+d).rotate();</span><br><span class="line">        Point pt3=Point(pt.x-d,pt.y).rotate();</span><br><span class="line">        Point pt4=Point(pt.x,pt.y-d).rotate();</span><br><span class="line">        <span class="keyword">int</span> p=lower_bound(Hx+<span class="number">1</span>,Hx+<span class="number">1</span>+mx,pt1.x)-Hx;    <span class="comment">//找到对应的set和vector</span></span><br><span class="line">        <span class="keyword">if</span>(Hx[p]==pt1.x) p1=gao(sx[p],vx[p],pt1.y,pt2.y);</span><br><span class="line">        p=lower_bound(Hx+<span class="number">1</span>,Hx+<span class="number">1</span>+mx,pt3.x)-Hx;</span><br><span class="line">        <span class="keyword">if</span>(Hx[p]==pt3.x) p2=gao(sx[p],vx[p],pt4.y,pt3.y);</span><br><span class="line">        p=lower_bound(Hy+<span class="number">1</span>,Hy+<span class="number">1</span>+my,pt2.y)-Hy;</span><br><span class="line">        <span class="keyword">if</span>(Hy[p]==pt2.y) p3=gao(sy[p],vy[p],pt3.x,pt2.x);</span><br><span class="line">        p=lower_bound(Hy+<span class="number">1</span>,Hy+<span class="number">1</span>+my,pt1.y)-Hy;</span><br><span class="line">        <span class="keyword">if</span>(Hy[p]==pt1.y) p4=gao(sy[p],vy[p],pt4.x,pt1.x);</span><br><span class="line">        <span class="keyword">if</span>(p1.SE&amp;&amp;p3.SE&amp;&amp;p1.SE==p3.SE) ans--;    <span class="comment">//区间端点去重</span></span><br><span class="line">        <span class="keyword">if</span>(p2.SE&amp;&amp;p3.FR&amp;&amp;p2.SE==p3.FR) ans--;</span><br><span class="line">        <span class="keyword">if</span>(p2.FR&amp;&amp;p4.FR&amp;&amp;p2.FR==p4.FR) ans--;</span><br><span class="line">        <span class="keyword">if</span>(p4.SE&amp;&amp;p1.FR&amp;&amp;p4.SE==p1.FR) ans--;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    n=read();A=read();B=read();</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=n;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        org[i].x=read();</span><br><span class="line">        org[i].y=read();</span><br><span class="line">        rtg[i]=org[i].rotate();</span><br><span class="line">        Hx[i]=rtg[i].x;</span><br><span class="line">        Hy[i]=rtg[i].y;</span><br><span class="line">    &#125;</span><br><span class="line">    d=<span class="built_in">abs</span>(org[A].x-org[B].x)+<span class="built_in">abs</span>(org[A].y-org[B].y);</span><br><span class="line">    prework();BFS();</span><br><span class="line">    <span class="built_in">printf</span>(<span class="string">"%lld\n"</span>,ans/<span class="number">2</span>);</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></div>

      
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